`f(x) = (((2x + 3)^2)(x - 2)^5)/((x^3)(x - 5)^2)` Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a...

`f(x) = (((2x + 3)^2)(x - 2)^5)/((x^3)(x - 5)^2)` Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum values.

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Textbook Question

Chapter 4, 4.6 - Problem 14 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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`f(x)=((2x+3)^2(x-2)^5)/(x^3(x-5)^2)`

Vertical asymptotes are the undefined points, also known as zeros of denominator.

Let's find the zeros of denominator of the function,

`x^3(x-5)^2=0`

`x^3=0 ,(x-5)^2=0`

`x=0, x=5`

Vertical Asymptotes are x=0 , x=5

For Horizontal Asymptotes

Degree of Numerator of the function=7

Degree of Denominator of the function=5

Degree of Numerator`>` 1+Degree of Denominator

`:.` There is no Horizontal Asymptote

Now let's find intercepts

x intercepts can be found when f(x)=0

`(2x+3)^2(x-2)^5=0`

`2x+3=0 , x-2=0`

`x=-3/2 , x=2`

So x intercepts are -1.5 and 2.

Since x is undefined at x=0 , there are no y intercepts.

See the attached graph and link.

From the graph,

Local MinimumĀ 

f(-1.5)=0

f(8) `~~` 600``

No Local Maximum

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